Because Maxpool function is a multivariate function, it is far more complex to be verified compared to Sigmoid/Tanh/Arctan function.
DeepPoly [1]} gives non-trivial linear bounds to Maxpooling function and its parallel work, CNN-Cert [2]} leverages a hyperplane containing n different particular points to give the upper and lower bound. Both these two state-of-the-art techniques fail to capture the characteristic of Maxpooling function and fail to give its Neuron-wise Tightest Linear Bounds, resulting in loose robustness bounds. To address this problem, we first leverage the notion of Neuron-wise Tightest and discover the Neuron-wise Tightest Linear Bounds:
[Neuron-wise Tightest Linear Bounds]

\(\forall k\in [K],i\in [n_k],m=\frac{l^{k-1}+u^{k-1}}{2}\) , if both \(u_i^k(m)\)  and \(-l_i^k(m)\)  reach the minimum, then the linear upper and lower linear approximations of every non-linear function are the Neuron-wise Tightest.

Due to the space limitation, the proof of Theorem REF  is deferred to Appendix.
By Theorem REF , we further propose Theorem REF  to help us find Neuron-wise Tightest Linear Bounds of Maxpooling function.
[Finding Neuron-wise Tightest Linear Bounds]

Given that \(f(x)\)  is a continuous function, then
